The Two-Dimensional Prouhet-Tarry-Escott Problem

In this paper we generalize the Prouhet–Tarry–Escott problem (PTE) to any dimension. The onedimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters n, k ∈ N, for two different multi-sets {(x1, y1), . . . , (xn, yn)}, {(x 1, y 1), . . . , (x n, y n)} of points from Z2 such that n i=1 xj i y d−j i = n i=1 x j i y d−j i for all d,j {0,...,k} with j d. We present parametric solutions for n {2, 3, 4, 6} with optimal size, i.e., with k = n − 1. We show that these solutions come from convex 2n-gons with all vertices in Z2 such that every line parallel to a side contains an even number of vertices and prove that such convex 2n-gons do not exist for other values of n. Furthermore we show that solutions to the two-dimensional PTE problem yield solutions to the one-dimensional PTE problem. Finally, we address the PTE problem over the Gaussian integers.      «

In this paper we generalize the Prouhet–Tarry–Escott problem (PTE) to any dimension. The onedimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters n, k ∈ N, for two different multi-sets {(x1, y1), . . . , (xn, yn)}, {(x 1, y 1), . . . , (x n, y n)} of points from Z2 such that n i=1 xj i y d−j i = n i=1 x j i y d−j i for all d,j {0,...,k} with j d. We present parametric solutions for n {2, 3, 4, 6} with optimal size,...     »